⟨2,X⟩≇⨁cyclic modules

hbghlyj

Show that $⟨2,X⟩$ is not $ℤ[X]$-linearly isomorphic to a direct sum of cyclic modules.

Czhang271828

Hint: If $M\subset A$ an additive subgroup is an (left/right) $A$-module, then $M$ is an (left/right) ideal of $A$ by definition.

(Here we need to prove $(X,2)$ is not isomorphic to the direct sum of some principal ideals....)

hbghlyj

MSE

The ideal $I=(2,X)$ of $\mathbb Z[X]$ is not a direct sum of (non-zero) cyclic $\mathbb Z[X]$-modules.

Let's suppose that $I=(2,X)$ is a direct sum of (non-zero) cyclic $\mathbb Z[X]$-modules. Then there exists a family $(N_{\alpha})_{\alpha\in A}$ of (non-zero) cyclic submodules of $I$ such that $I=\sum_{\alpha\in A}N_{\alpha}$, and $N_{\beta}\cap\sum_{\alpha\ne\beta}N_{\alpha}=0$ for all $\beta\in A$. In particular, $N_{\alpha}$ are principal ideals in $\mathbb Z[X]$. But $N_{\alpha}\cap N_{\beta}\ne 0$ for $\alpha\ne\beta$ (if $x_{\alpha}\in N_{\alpha}$ and $x_{\beta}\in N_{\beta}$, then $x_{\alpha}x_{\beta}\in N_{\alpha}\cap N_{\beta}$). This leads us to the conclusion that $|A|=1$, that is, $I$ is principal, a contradiction

hbghlyj

I think the main point of 3# is "Two principal ideals of $\Bbb Z[X]$ are not disjoint so not isomorphic to direct sum"